Finite Group Databasebeta

← #223663

Subgroups — #223663

Order 512.

859 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36193C211yesmaximal, characteristic
2#22601C211yesmaximal, characteristic
3#13893C211yesmaximal, characteristic
4#9912C211yesmaximal, characteristic
5#9888C211yesmaximal, characteristic
6#9153C211yesmaximal, characteristic
7#9144C211yesmaximal, characteristic
8#9116C211yesmaximal, characteristic
9#9112C211yesmaximal, characteristic
10#8717C211yesmaximal, characteristic
11#8716C211yesmaximal, characteristic
12#8576C211yesmaximal, characteristic
13#8217C211yesmaximal, characteristic
14#8018C211yesmaximal, characteristic
15#8014C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18#2857C2211yescharacteristic
19#2855C2211yescharacteristic
20#2855C2211yescharacteristic
21#2567C411yescharacteristic, complemented
22#2552C411yescharacteristic, complemented
23#2520—21no—
24#2520—21no—
25#2511C411yescharacteristic, complemented
26#2511C411yescharacteristic, complemented
27#2508—21no—
28#2507C411yescharacteristic, complemented
29#2507C411yescharacteristic, complemented
30#2442—21no—
31#2438—21no—
32#2438—21no—
33#2370—21no—
34#2369C411yescharacteristic, complemented
35#2355C411yescharacteristic, complemented
36#2344—21no—
37C4×C22⋊Q8C2211yescharacteristic
38#2285C2211yescharacteristic
39#2285C2211yescharacteristic
40C2×#383C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43#1940—21no—
44#1917—21no—
45#1916—21no—
46#1817C2211yescharacteristic
47#1817C2211yescharacteristic
48#1797C2211yescharacteristic
49#1771—21no—
50#1749C2211yescharacteristic
51C2×#328C2211yescharacteristic
52C2×#328C2211yescharacteristic
53C2×#328C2211yescharacteristic
54C2×#328C2211yescharacteristic
55#1442—21no—
56#1441—21no—
57#1436—21no—
58#1434—21no—
59#1434C2211yescharacteristic
60#1434C2211yescharacteristic
61#1434C2211yescharacteristic
62#1434C2211yescharacteristic
63#1294C2211yescharacteristic
64#1294C2211yescharacteristic
65#1294C2211yescharacteristic
66#1294C2211yescharacteristic
67#1294C2211yescharacteristic
68#1294C2211yescharacteristic
69#1294C2211yescharacteristic
70#1294C2211yescharacteristic
71#1294C2211yescharacteristic
72#1294C2211yescharacteristic
73#1294C2211yescharacteristic
74#1294C2211yescharacteristic
75C2×C4⋊Q8C2×C411yescharacteristic
76C2×#125—21no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8C2311yescharacteristic
79C2×C4×Q8C2×C411yescharacteristic
80C2×#117—21no—
81C2×#117—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4D411yescharacteristic
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#432—22no—
90#432—41no—
91#431—22no—
92#431—41no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C22⋊C8C2311yescharacteristic
97C2×C22⋊C8C2311yescharacteristic
98C2×C22⋊C8C2311yescharacteristic
99C2×C22⋊C8C2311yescharacteristic
100C2×#97—21no—
101C2×#97—21no—
102C2×C4×C8—21no—
103#400—21no—
104#400—21no—
105#400—21no—
106#400—21no—
107#398—21no—
108#398—21no—
109#398—21no—
110#398—21no—
111#398—21no—
112#398—21no—
113#395—21no—
114#395—21no—
115#391—21no—
116#391—21no—
117#391—21no—
118#391C2×C411yescharacteristic
119#391C2×C411yescharacteristic
120#391C2×C411yescharacteristic
121#391C2×C411yescharacteristic
122#391C2×C411yescharacteristic
123#389—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#389C2×C411yescharacteristic
131#389C2×C411yescharacteristic
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#389C2×C411yescharacteristic
136#389C2×C411yescharacteristic
137#387—21no—
138#387—21no—
139#387—21no—
140#387—21no—
141#387—21no—
142#387—21no—
143#387—21no—
144#387—21no—
145#387—21no—
146#387—21no—
147#387—21no—
148#387—21no—
149#387—21no—
150#385—21no—
151#385—21no—
152#385—21no—
153#384C2×C411yescharacteristic
154#384C2×C411yescharacteristic
155#384C2×C411yescharacteristic
156#384C2×C411yescharacteristic
157#384C2×C411yescharacteristic
158#383—22no—
159#383—41no—
160#382—21no—
161#382—21no—
162#382—21no—
163#381—21no—
164#381—21no—
165#380—21no—
166#380—21no—
167#380—21no—
168#380C2311yescharacteristic
169#380C2311yescharacteristic
170#380C2311yescharacteristic
171C4×C4⋊C4—21no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4—21no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—41no—
178C4×C4⋊C4C2×C411yescharacteristic
179C4×C4⋊C4C2×C411yescharacteristic
180C4×C4⋊C4C2×C411yescharacteristic
181C4×C4⋊C4C2×C411yescharacteristic
182C4×C4⋊C4C2×C411yescharacteristic
183C4×C4⋊C4C2×C411yescharacteristic
184C4×C4⋊C4C2×C411yescharacteristic
185C4×C4⋊C4C2×C411yescharacteristic
186C4×C4⋊C4C2×C411yescharacteristic
187C4×C4⋊C4D411yescharacteristic
188C4×C4⋊C4D411yescharacteristic
189C4×C4⋊C4D411yescharacteristic
190C4×C4⋊C4D411yescharacteristic
191C4×C22⋊C4—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95D411yescharacteristic
203C2×#95D411yescharacteristic
204C2×#95C2311yescharacteristic
205C2×#95C2311yescharacteristic
206C2×#95C2311yescharacteristic
207C2×#95C2311yescharacteristic
208#336—21no—
209#336—21no—
210#336—21no—
211#336—21no—
212#336—21no—
213#336—21no—
214#328—22no—
215#328—22no—
216#328—41no—
217#328—41no—
218C22×Q8—21no—
219C22×Q8C22×C411yescharacteristic
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2411yescharacteristic
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C4⋊Q8—42no—
236#125—44no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C22⋊Q8—81no—
242C22⋊Q8—81no—
243C4×Q8—42no—
244#117—81no—
245#117—81no—
246#117—81no—
247#117—81no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4C22⋊C411yescharacteristic
275C2×C4⋊C4C22⋊C411yescharacteristic
276C2×C4⋊C4C22⋊C411yescharacteristic
277C2×C4⋊C4C22⋊C411yescharacteristic
278C2×C4⋊C4C22⋊C411yescharacteristic
279C2×C4⋊C4C22⋊C411yescharacteristic
280C2×C4⋊C4C22⋊C411yescharacteristic
281C2×C4⋊C4C22⋊C411yescharacteristic
282C2×C4⋊C4C22×C411yescharacteristic
283C2×C4⋊C4C22×C411yescharacteristic
284C2×C4⋊C4C22×C411yescharacteristic
285C2×C4⋊C4C22×C411yescharacteristic
286C2×C4⋊C4C22×C411yescharacteristic
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C4⋊C4C22×C411yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42—81no—
311C2×C42C22×C411yescharacteristic
312C2×C42C22×C411yescharacteristic
313C2×C42C22×C411yescharacteristic
314C2×C42C22×C411yescharacteristic
315C2×C42C22×C411yescharacteristic
316C2×C42C22×C411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C4⋊C8—42no—
324C4⋊C8—42no—
325C4⋊C8—42no—
326C22⋊C8—21no—
327C22⋊C8—21no—
328C22⋊C8—21no—
329C22⋊C8—21no—
330C22⋊C8—21no—
331C22⋊C8—21no—
332C22⋊C8—21no—
333C22⋊C8#4112yes—
334#97—42no—
335#97—42no—
336C4×C8—42no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—42no—
370#95—42no—
371#95—42no—
372#95—42no—
373#95—42no—
374#95—42no—
375#95—42no—
376#95—42no—
377#95—81no—
378#95—81no—
379#95C22⋊C411yescharacteristic
380#95C22⋊C411yescharacteristic
381#95C22⋊C411yescharacteristic
382#95C22⋊C411yescharacteristic
383#95C22⋊C411yescharacteristic
384#95C22⋊C411yescharacteristic
385#95C22⋊C411yescharacteristic
386#95C22⋊C411yescharacteristic
387#95#4112yes—
388C24C22×D411yescharacteristic
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—42no—
392C2×Q8—41no—
393C2×Q8—41no—
394C2×Q8—81no—
395C2×Q8—81no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4C2×C22⋊C411yescharacteristic
461C22×C4C2×C22⋊C411yescharacteristic
462C22×C4C2×C22⋊C411yescharacteristic
463C22×C4C2×C22⋊C411yescharacteristic
464C22×C4C2×C22⋊C411yescharacteristic
465C22×C4C2×C22⋊C411yescharacteristic
466C22×C4C2×C22⋊C411yescharacteristic
467C22×C4C2×C22⋊C411yescharacteristic
468C22×C4C2×C22⋊C411yescharacteristic
469C22×C4C2×C22⋊C411yescharacteristic
470C22×C4C2×C22⋊C411yescharacteristic
471C22×C4C2×C22⋊C411yescharacteristic
472C22×C4#13712yes—
473C22×C4#13712yes—
474C22×C4C23×C411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C2×#4111yescharacteristic
477C22×C4C2×#4111yescharacteristic
478C22×C4H(5,𝔽2)11yescharacteristic
479C22×C42−1+411yescharacteristic
480C2×C8—22no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—41no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—48no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—81no—
519C4⋊C4—81no—
520C4⋊C4—81no—
521C4⋊C4—81no—
522C4⋊C4—81no—
523C4⋊C4—81no—
524C4⋊C4—81no—
525C4⋊C4—81no—
526C4⋊C4—81no—
527C4⋊C4—81no—
528C4⋊C4—81no—
529C4⋊C4—81no—
530C22⋊C4—41no—
531C22⋊C4—41no—
532C22⋊C4—41no—
533C22⋊C4—41no—
534C22⋊C4—41no—
535C22⋊C4—41no—
536C22⋊C4—41no—
537C22⋊C4—41no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—22no—
555C42—22no—
556C42—41no—
557C42—41no—
558C42—42no—
559C42—42no—
560C42—81no—
561C42—81no—
562C42—81no—
563C42—81no—
564C42—84no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23C22×C22⋊C411yescharacteristic
570C23#54611yescharacteristic
571C23#54711yescharacteristic
572C23C2×#13711yescharacteristic
573C23C2×#13711yescharacteristic
574C23#57611yescharacteristic
575C23#57711yescharacteristic
576Q8—81no—
577Q8—81no—
578Q8—162no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—42no—
661C2×C4—81no—
662C2×C4—81no—
663C2×C4—81no—
664C2×C4—81no—
665C2×C4—81no—
666C2×C4—81no—
667C2×C4—81no—
668C2×C4—81no—
669C2×C4—81no—
670C2×C4—81no—
671C2×C4—81no—
672C2×C4—81no—
673C2×C4—81no—
674C2×C4—81no—
675C2×C4—81no—
676C2×C4—81no—
677C2×C4—81no—
678C2×C4—81no—
679C2×C4—81no—
680C2×C4—81no—
681C2×C4—81no—
682C2×C4—81no—
683C2×C4—81no—
684C2×C4—81no—
685C2×C4—81no—
686C2×C4—81no—
687C2×C4#41012yes—
688C2×C4#41912yes—
689C2×C4#41912yes—
690C2×C4#42112yes—
691C2×C4C4×#4111yescharacteristic
692C2×C4#52011yescharacteristic
693C2×C4#54611yescharacteristic
694C2×C4#54711yescharacteristic
695C8—42no—
696C8—81no—
697C8—81no—
698C8—81no—
699C22—21no—
700C22—21no—
701C22—21no—
702C22—21no—
703C22—21no—
704C22—21no—
705C22—21no—
706C22—21no—
707C22—21no—
708C22—21no—
709C22—21no—
710C22—21no—
711C22#233911yescharacteristic
712C22C2×#41011yescharacteristic
713C22#288411yescharacteristic
714C22C2×#41911yescharacteristic
715C22#289711yescharacteristic
716C22C2×#42111yescharacteristic
717C22#290111yescharacteristic
718C22#331411yescharacteristic
719C22#331611yescharacteristic
720C22#334411yescharacteristic
721C22#334611yescharacteristic
722C4—22no—
723C4—21no—
724C4—21no—
725C4—21no—
726C4—21no—
727C4—22no—
728C4—22no—
729C4—22no—
730C4—41no—
731C4—41no—
732C4—41no—
733C4—41no—
734C4—41no—
735C4—41no—
736C4—41no—
737C4—41no—
738C4—82no—
739C4—82no—
740C4—81no—
741C4—81no—
742C4—81no—
743C4—81no—
744C4—81no—
745C4—81no—
746C4—81no—
747C4—81no—
748C4—161no—
749C4—161no—
750C2—21no—
751C2—21no—
752C2—21no—
753C2—21no—
754C2#1756211yescharacteristic
755C2#1760211yescharacteristic
756C2#1760711yescharacteristic
757C2#1763211yescharacteristic
758C2#2247811yescharacteristic
759C2#2249711yescharacteristic
760C2#2250611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
21. #2567C4semidirect
22. #2552C4semidirect
25. #2511C4semidirect
26. #2511C4semidirect
28. #2507C4semidirect
29. #2507C4semidirect
34. #2369C4semidirect
35. #2355C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#361932
#226012
#138932
#99122
#98882
#91532
#91442
#91162
#91122
#87172
#87162
#85762
#82172
#80182
#80142